It depends. It may be possible to take half a credit for a dependent as your wife and her sister wish to do, but typically there is a requirement to provide more than 50% of the relative's support to qualify as a dependent.
By caregiver/disability transfer, if you are referring to the medicaid state plans where family members are paid to care for their loved one, through a self-directed personal assistance service state plan option, then there could be a more objective way of determining the caregiving provided by each family member for their mother.
Click the links for details:
Steps to claiming an elderly parent as a dependent
IRS Form 2120 Multiple Support Declaration
Medical Expenses Checklist
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